Shadow Trajectory Model for Fast Low-Thrust Indirect Optimization

Preliminary design of low-thrust trajectories generally benefits from broad searches over the feasible space. Despite convergence issues, indirect methods are generally faster than direct methods and are therefore well-suited for such searches. Nonetheless, indirect solutions typically require expensive numerical integration of at least the state and costate equations. Here, based on the physical interpretation of the primer vector, a fast model that approximates solutions to all the dynamics is introduced. If the ballistic dynamics have a closed-form solution, then the costate equations can be approximated without resorting to numerical integration. Analogous to the Sims–Flanagan model for direct optimization, the new model approximates thrust arcs with a series of ballistic arcs and impulsive maneuvers. The closed-form solution is obtained using a Sundman-transformed independent variable, which also provides an efficient discretization. Furthermore, a low-order series solution is used for the ballistic propagation. The model is introduced and evaluated for speed and accuracy using examples with Keplerian dynamics. Speedups vary according to thrust and discretization levels. For accuracies relevant to preliminary design, order-of-magnitude speedups are achieved.